Bounds for the number of multidimensional partitions

We obtain estimates for the number $p_d\left(n\right)$ of $(d-1)$-dimensional integer partitions of a number $n$. It is known that the two-sided inequality $C_1\left(d\right)n^{1-1/d}<log{p}_d\left(n\right)<C_2\left(d\right)n^{1-1/d}$ is always true and that $C_1\left(d\right)>1$ whenever $logn>3d$. However, establishing the $“$right$”$ dependence of $C_2$ on $d$ remained an open problem. We show that if $d$ is sufficiently small with respect to $n$, then $C_2$ does not depend on $d$, which means that $log{p}_d\left(n\right)$ is up to an absolute constant equal to $n^{1-1/d}$. Besides, we provide estimates of $p_d\left(n\right)$ for different ranges of $d$ in terms of $n$, which give the asymptotics of $log{p}_d\left(n\right)$ in each case.